# Recent Activity

## Are there an infinite number of lucky primes? ★

Author(s): Lazarus: Gardiner: Metropolis; Ulam

Conjecture   If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.

Keywords: lucky; prime; seive

## Something like Picard for 1-forms ★★

Author(s): Elsner

Conjecture   Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .

## The robustness of the tensor product ★★★

Author(s): Ben-Sasson; Sudan

Problem   Given two codes , their Tensor Product is the code that consists of the matrices whose rows are codewords of and whose columns are codewords of . The product is said to be robust if whenever a matrix is far from , the rows (columns) of are far from (, respectively).

The problem is to give a characterization of the pairs whose tensor product is robust.

Keywords: codes; coding; locally testable; robustness

## Schanuel's Conjecture ★★★★

Author(s): Schanuel

Conjecture   Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .

Keywords: algebraic independence

## Beneš Conjecture ★★★

Author(s): Beneš

Given a partition of a finite set , stabilizer of , denoted , is the group formed by all permutations of preserving each block in .

Problem  ()   Find a sufficient condition for a sequence of partitions of to be universal, i.e. to yield the following decomposition of the symmetric group on : In particular, what about the sequence , where is a permutation of ?
Conjecture  (Beneš)   Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then

Keywords:

## Frankl's union-closed sets conjecture ★★

Author(s): Frankl

Conjecture   Let be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of .

Keywords:

## Double-critical graph conjecture ★★

Author(s): Erdos; Lovasz

A connected simple graph is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.

Conjecture   is the only -chromatic double-critical graph

Keywords: coloring; complete graph

## Shuffle-Exchange Conjecture ★★★

Author(s): Beneš; Folklore; Stone

Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as , where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.

Problem  (SE)   Find .
Conjecture  (SE)   .

Keywords:

## Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.

Conjecture   If is the maximal degree of a graph , then is strongly -colorable.

Keywords: strong coloring

## Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every , all but finitely many -regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

## Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere? ★★★

Author(s): Novikov

Problem   Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?

Keywords: 4-sphere; algorithm

## What is the homotopy type of the group of diffeomorphisms of the 4-sphere? ★★★★

Author(s): Smale

Problem   has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .

Keywords: 4-sphere; diffeomorphisms

## Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere? ★★★

Author(s): Kirby

Problem   Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.

Keywords: 3-manifold; 4-sphere; embedding

## Fundamental group torsion for subsets of Euclidean 3-space ★★

Author(s): Ancient/folklore

Problem   Does there exist a subset of such that its fundamental group has an element of finite order?

Keywords: subsets of euclidean space; torsion

## Which homology 3-spheres bound homology 4-balls? ★★★★

Author(s): Ancient/folklore

Problem   Is there a complete and computable set of invariants that can determine which (rational) homology -spheres bound (rational) homology -balls?

Keywords: cobordism; homology ball; homology sphere

## Realisation problem for the space of knots in the 3-sphere ★★

Author(s): Budney

Problem   Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .

Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .

There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .

Problem: What representations can be obtained?

Keywords: knot space; symmetry

## Slice-ribbon problem ★★★★

Author(s): Fox

Conjecture   Given a knot in which is slice, is it a ribbon knot?

Keywords: cobordism; knot; ribbon; slice

## Smooth 4-dimensional Poincare conjecture ★★★★

Author(s): Poincare; Smale; Stallings

Conjecture   If a -manifold has the homotopy type of the -sphere , is it diffeomorphic to ?

Keywords: 4-manifold; poincare; sphere

## Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

Problem   Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

## Are different notions of the crossing number the same? ★★★

Author(s): Pach; Tóth

Problem   Does the following equality hold for every graph ?

The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number , we minimize the number of pairs of edges that cross.

Keywords: crossing number; pair-crossing number